Some function classes closed under infimal convolution

• DOI: 10.1007/bf00933523
• Publication Date: 2005-01-01T17:38:12Z
• Authors: D. H. Martin
• Keywords: 0101 mathematics; 01 natural sciences

It is shown that the following classes of functionsf, each defined on some subsetD of a fixed Hausdorff topological vector spaceE, are closed under the binary operation of infimal convolution: (a) the class of functionsf:D→[−∞, ∞) having connectedstrict lower level sets; (b) the class of functionsf:D→ℝ having compact connected lower level sets; and (c) the class of functionsf:D→ℝ having compact lower level sets and for which every local minimizer is global. In (a),E need not be Hausdorff; while in (a) and (b), the word “connected” may be replaced by the word “path-connected.”